How recent is the use of "so" as in "I'm so full", and did it originate in US or UK English?

It is so old. It is also so '80s-to-present.

There is a usage of so that has become far more popular in the last 40 years, but "I'm so full" would not be an example of it. "So" has applied to adjectives since Old English: so old, so bold, so cold, so beautiful.

The recent usage involves applying so as an intensifier to nouns, other adverbs, verbs, or adjectives that customarily would not take an intensifier. Here's how the Oxford English Dictionary describes it:

a. Modifying a noun, or an adjective or adverb which does not usually admit comparison: extremely, characteristically.

b. Modifying a verb: definitely, decidedly. Frequently in negative constructions.

The Routledge Dictionary of Modern American Slang and Unconventional English, noting the "attitude" and "pronunciation" with which this meaning is delivered, dates this usage to 1979 as used in the film Manhattan:

Oh, please, you know, God, you're so the opposite!

"So" didn't usually modify nouns before this point. Here's an example of so applied to the colloquial phrase "into you":

He's made it clear that he's so not into you that he couldn't even bother to leave you a Post-it.

As for region of origin, the Oxford English Dictionary claims it is "chiefly US," and the first cited instance is from an American film.

The OED has citations for that usage from old English until present. I think the problem is in intonation. If you imagine that the "so" with a sarcastic eye-roll, meaning "not at all" then I suspect, yes, the usage originated in California (specifically valley speak) and is relatively (80s-90s) recent.

Edit you want citations:

Shakespere's much ado about nothing has: "That you haue such a Februarie face, So full of frost, of storme, and clowdinesse."

And here's another public domain citation from British English in 1819.

And yet another example from the King James Bible, published 1611:

Genesis 27:20: And Isaac said unto his son, How is it that thou hast found it so quickly, my son?

Those answer the OPs question, it's English usage from before America was a colony, let alone a country. As to the speculation as to why some other site thinks this is a modern Californian invention, without more information, the answer can only be speculation.